Two Approaches to Non-Commutative Geometry

TL;DR

This paper explores two historical approaches to geometry—algebraic and group-theoretic—and discusses their modern interpretations in non-commutative geometry, highlighting their interconnectedness and relevance to quantum mechanics.

Contribution

It provides a comparative analysis of algebraic and group-theoretic methods in non-commutative geometry, emphasizing their historical roots and modern applications.

Findings

Algebraic and group-theoretic approaches are interconnected through Galois.

Modern non-commutative geometry bridges these approaches with quantum mechanics.

Connections between various mathematical objects are discussed in the context of non-commutative geometry.

Abstract

Looking to the history of mathematics one could find out two outer approaches to Geometry. First one (algebraic) is due to Descartes and second one (group-theoretic)--to Klein. We will see that they are not rivalling but are tied (by Galois). We also examine their modern life as philosophies of non-commutative geometry. Connections between different objects (see keywords) are discussed. Keywords: Heisenberg group, Weyl commutation relation, Manin plain, quantum groups, SL(2, R), Hardy space, Bergman space, Segal-Bargmann space, Szeg"o projection, Bergman projection, Clifford analysis, Cauchy-Riemann-Dirac operator, Moebius transformations, functional calculus, Weyl calculus (quantization), Berezin quantization, Wick ordering, quantum mechanics.